In Cox's theorem, probability is taken as a primitive ( that is, not further analyzed ) and the emphasis is on constructing a consistent assignment of probability values to propositions.

An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem.

* Cox's theorem

Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates.

As the laws of probability derived by Cox's theorem are applicable to any proposition, logical probability is a type of Bayesian probability.

Cox's theorem has come to be used as one of the justifications for the

The original formulation of Cox's theorem is in, which is extended with additional results and more discussion in.

Richard Threlkeld Cox ( 1898 – May 2, 1991 ) was a professor of physics at Johns Hopkins University, known for Cox's theorem relating to the foundations of probability.

Richard Cox's most important work was Cox's theorem.

" Richard Threlkeld Cox later showed in Cox's theorem that any extension of Aristotelian logic to incorporate truth values between 0 and 1, in order to be consistent, must be equivalent to Bayesian probability.

# REDIRECT Cox's theorem

Cox's Braves did not advance past the first round of the playoffs in any of their last five appearances.

Cox's absence is never addressed in this movie, nor is he mentioned at any point.

As of recently, WKMG is the last " Big Six " affiliate in Orlando that is not part of any television duopoly ( Fox / Newscorp's WOFL / WRBW, Cox's WFTV / WRDQ and Hearst-Argyle's WESH / WKCF ).

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

This implies, by the Bolzano – Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

The three classes are groups of GF ( 2 ) type ( classified mainly by Timmesfeld ), groups of " standard type " for some odd prime ( classified by the Gilman – Griess theorem and work by several others ), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.

For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:

In fact, Cantor's method of proof of this theorem implies the existence of an " infinity of infinities ".

Together with soundness ( whose verification is easy ), this theorem implies that a formula is logically valid if and only if it is the conclusion of a formal deduction.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

The Paley – Wiener theorem immediately implies that if f is a nonzero distribution of compact support ( these include functions of compact support ), then its Fourier transform is never compactly supported.

Shannon's theorem also implies that no lossless compression scheme can compress all messages.

Bell's theorem implies, and it has been proven mathematically, that compatible measurements cannot show Bell-like correlations, and thus entanglement is a fundamentally non-classical phenomenon.

One might expect that by the Hahn-Banach theorem for bounded linear functionals, every bounded linear functional on C < sub > c </ sub >( X ) extends in exactly one way to a bounded linear functional on C < sub > 0 </ sub >( X ), the latter being the closure of C < sub > c </ sub >( X ) in the supremum norm, and that for this reason the first statement implies the second.

The Heine – Borel theorem implies that a Euclidean n-sphere is compact.

The integrability condition and Stokes ' theorem implies that the value of the line integral connecting two points is independent of the path.

If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that

The Arzelà – Ascoli theorem implies that if is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence.

For instance, Bell's theorem implies that quantum mechanics cannot satisfy both local realism and counterfactual definiteness.

Noether's theorem implies that there is a conserved current associated with translations through space and time.

However, if the errors are not normally distributed, a central limit theorem often nonetheless implies that the parameter estimates will be approximately normally distributed so long as the sample is reasonably large.

In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice.

To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty.

Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

The original Nernst heat theorem makes the weaker and less controversial claim that the entropy change for any isothermal process approaches zero as T → 0:

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any object point unite in an image point ; and therefore an object space is reproduced in an image space.

Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

According to the theorem, it is possible to expand any power of x + y into a sum of the form

The binomial theorem can be applied to the powers of any binomial.

Schaefer's dichotomy theorem states that, for any restriction to Boolean operators that can be used to form these subformulae, the corresponding satisfiability problem is in P or NP-complete.

Again from the Heine – Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact.

* ( Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation ).

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

As a consequence of the Nyquist – Shannon sampling theorem, any spatial waveform that can be displayed must consist of at least two pixels, which is proportional to image resolution.

" From these principles and some additional constraints —( 1a ) a lower bound on the linear dimensions of any of the parts, ( 1b ) an upper bound on speed of propagation ( the velocity of light ), ( 2 ) discrete progress of the machine, and ( 3 ) deterministic behavior — he produces a theorem that " What can be calculated by a device satisfying principles I – IV is computable.

Owing respectively to Green's theorem and the divergence theorem, such a field is necessarily conserved and free from sources or sinks, having net flux equal to zero through any open domain.

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free ( and hence has only one prime factor of two ) will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.

This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.